take the following integral
[tex]\binom{m+n+1}{m,n}\int_0^1 u^{m} (1-u)^n du[/tex]
consider the integral as a sum, we see that given m+n+1 particles on (0,1), it sums the probability of finding m particles in (0, u), 1 particle around the point u and n particles in (u, 1), so it must be one. This gives the beta integral for integers m, n!!!
[tex]\int_0^1 u^{m} (1-u)^n du=\frac{m!n!}{(m+n+1)!}[/tex]